Mistake Master
Convert first, compute second
The arithmetic does not care that the rate was per hour and the time was in minutes, or that the axis said "in thousands." It multiplies whatever it is fed and produces a confident, spectacular wrong answer. Every one of those answers is pre-written into the choices. This lesson builds the convert-first reflex and the read-the-label habit that starve them.
§1
What this topic is about
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Numbers on the SAT arrive wearing labels: hours, centimeters, "in thousands," percent, per serving. The arithmetic is indifferent to the labels, and that is the danger. This topic trains two habits: match units before computing, and apply every scale label at the moment of reading.
§2
Convert before the arithmetic
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A per-hour rate cannot multiply minutes; meters cannot divide centimeters. The number that comes out of mismatched units is always wrong and usually spectacular, which makes it easy to catch and embarrassing to miss.
- Convert everything to ONE unit before any multiplication or division.
- Rates name their time unit: per hour means hours must enter, not minutes.
- Carry the units through the arithmetic; the answer's units should assemble themselves.
Worked example. A car travels at $30$ miles per hour for $90$ minutes. How far does it go?
$90$ minutes is $1.5$ hours.
$30 \cdot 1.5 = 45$ miles. The raw product $30 \cdot 90 = 2{,}700$ carries units of mile-minutes per hour, which is not a distance.
§3
The scale label is part of the number
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An axis marked "in thousands" multiplies every bar by $1{,}000$. A percent divides by $100$ on its way into any computation. A recipe's amount belongs to the whole batch, not to one serving. In each case the label is half the value.
- Apply the axis scale AT THE READ, before the number travels anywhere.
- Percent converts at the door: $5\%$ computes as $0.05$.
- Batch quantities: find what ONE unit is worth, then scale to the asked count.
Worked example. A recipe serving $4$ uses $300$ grams of rice. How much for $10$ people?
$300 \div 4 = 75$ grams per person.
$75 \cdot 10 = 750$ grams. Multiplying $300 \cdot 10$ treats the batch amount as a per-person amount.
§4
Stacked scales
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The hardest items stack two conversions: a scaled axis AND a price per unit, a unit conversion AND a rate. Run them one at a time, each with its units named.
- One conversion per line; never two in one mental step.
- After each line, say what the number now counts.
- A sanity check at the end: is $2{,}700$ miles reasonable for a $90$-minute drive?
Worked example. An axis reads "tickets, in hundreds"; Saturday's bar reaches $24$; tickets cost $\$5$. Revenue?
$24$ hundreds is $2{,}400$ tickets.
$2{,}400 \cdot 5 = \$12{,}000$. Skipping the scale gives $\$120$, a suspiciously small Saturday.
§5
The units reread
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Like the ask-reread of Topic 6.1, the units reread happens at the grid: does the number in your hand carry the units the question named, at the scale it named them?
Convert to one unit before computing, apply axis scales and percent conversions at the moment of reading, unstack multiple scales one line at a time, and reread the asked units before gridding.
§6
Two patterns that cost real points
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Two patterns cover this topic. They are the same ones the diagnostic routes on.
Mismatched units run through the arithmetic.
A per-hour rate meets minutes, meters divide centimeters, and the computation is flawless on numbers that never spoke the same unit.
Fix. Convert first, compute second. Carry the units through every step and let them assemble the answer's label; a label that does not match the ask convicts the setup.
A scale label goes unapplied.
The bar height travels without its "in thousands," the percent multiplies raw, the batch amount serves one person.
Fix. Treat the label as part of the number from the first read. Bars get their multiplier immediately, percents convert at the door, and batch amounts get divided down to one unit before scaling up.
Ten quick checks across the two patterns: rate-time mismatches, length conversions, axis scales, percent conversion, batch-to-serving scaling, and stacked scales. Pick or type your answer, then check. Progress is saved.